Fast proximal algorithms for applications in viscoplasticity.
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
Numerical flow simulations for viscoplastic fluids have posed, and continue to pose major challenges. The large scale of industrially relevant flow problems coupled with the highly nonlinear and nonsmooth nature of viscoplastic materials still poses too high an obstacle even for modern computer clusters. This research aims to provide more efficient numerical schemes for flow simulations of Bingham, Casson and Herschel-Bulkley fluids without perturbing their viscoplastic behaviour by smoothing or regularisation. Two main contributions form the focus of this thesis: firstly, a new dual formulation of such problems and secondly, their numerical solution by proximal gradient or proximal Newton-type methods. To this end, we initially study a class of generic convex optimisation problems in Hilbert spaces. We design dual-based algorithms in the appropriate function spaces and derive properties of the primal problem that guarantee their applicability and convergence. ‘Fast’ or ‘accelerated’ proximal gradient methods can be adapted to viscoplastic flow problems, to yield strong convergence of order O(1=k), as the iteration counter k ! 1. This contrasts to O(1= p k) convergence of state-of-the-art solvers in viscoplasticity. Accelerated second-order methods of Newton type are particularly advantageous for resolving the additional nonlinearity that arises in Casson and Herschel-Bulkley flow problems. We observe that these algorithms can converge several times faster than classical alternatives. Simulations of stationary and time-dependent flows through pipe cross-sections and two-dimensional cavities demonstrate the viability and efficiency of this approach. One may anticipate that these new numerical methods bring us an important step closer towards the industrial applicability of computational viscoplasticity.